Multiplicity existence for sublinear fractional Laplacian problems

Abstract

The purpose of this paper is mainly to investigate the existence of weak solutions of fractional Laplacian equations with sublinear growth and oscillatory behavior. By using variational methods and abstract critical point theory, we want to establish the existence of multiplicity positive, multiplicity negative, and in particular, of multiplicity sign changing solutions. The multiplicity results depend on the real parameter . The key point is the choice of the framework to study the existence of weak solutions. In the suitable framework, by verifying that the conditions in abstract critical point theory are satisfied, we are able to fulfill our strategy.

Keywords:

• Fractional Laplacian problems
• variational methods
• multiple solutions

AMS Subject Classifications:

• 35R11
• 35A15
• 35J60
• 47G20

Notes

No potential conflict of interest was reported by the authors.

Additional information

Funding

The first author was supported by the National Natural Science Foundation of China [grant number 11371110]; this work was done while the first author was visiting the Dipartimento di Matematica e Informatica, University of Perugia. Y. Fu is grateful for the hospitality and support received during his research there throughout June and July 2016. The second author is a member of the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). The manuscript was realized within the auspices of the INdAM – GNAMPA Project Problemi variazionali su varietà Riemanniane e gruppi di Carnot [Prot_2016_000421]. P. Pucci was partly supported by the Italian MIUR project Variational and perturbative aspects of nonlinear differential problems [201274FYK7].